Correct output chi-square values and p-values by genomic control.

This somewhat older method, pioneered by Devlin and Roeder [Devlin and Roeder 1999]¹, notes that the chi-squared distribution of statistics from association tests being confounded by stratification will be more “spread out” than it should be. This will result in its median being higher than the median of a true chi-square distribution. Several models exist for how much the distribution should be spread out, depending on the test type, but the bottom line is that the distribution will usually be uniformly spread out by a certain “inflation factor” λ.

The technique of Genomic Control measures this “inflation factor” λ by taking the median of the distribution of the chi-square statistic from results of an actual test done over a set of markers from the study in question, and dividing this median by the median of the corresponding (ideal) chi-square distribution. If the result is less than one, the distribution is considered close enough to ideal and λ is taken to be one.

Then, Genomic Control applies its correction by dividing the actual association test chi-square statistic results by this λ, thus possibly making these results appropriately more pessimistic.

Two approaches exist for this:

• (Appropriate for studies over a small number of markers:) Measure the “inflation factor” λ over a set of markers designed to indicate population stratification. Then use this λ on the actual association test (presumably done for just a few candidate markers).
• (Appropriate for whole-genome scans or a large number of markers:) Measure the “inflation factor” λ over the actual association tests being done. Then afterward, use this λ on all chi-square results so obtained.

HelixTree facilitates both approaches.

Select Show Inflation Factor (Lambda), Chi-Squares, and Corrected Values to find inflation factors (λ) and the results of applying the Genomic Control technique on chi-squares, p-values, Bonferroni-adjusted p-values, and False Discovery Rates.

Select Correct Using This Inflation Factor (Lambda) Instead: and enter a λ value to use an “inflation factor” that was determined from a previous association test run or other previous data.

NOTE: The inflation factor relates to the chi-square statistic. After a chi-square statistic has been corrected through Genomic Control, the normal procedure for finding the approximate p-value is still followed. If there had been “inflation”, the GC-corrected p-value will be pushed up closer to one.

References

1. B. Devlin, Kathryn Roeder, ’Genomic Control for Association Studies’, Biometrics, Vol. 55, No. 4 (Dec., 1999), pp. 997-1004